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FUNDAMENTA
MATHEMATICAE
159 (1999)
The concept of boundedness and the Bohr
compactification of a MAP Abelian group
by
Jorge G a l i n d o and Salvador H e r n á n d e z (Castellón)
Abstract. Let G be a maximally almost periodic (MAP) Abelian group and let B be a
boundedness on G in the sense of Vilenkin. We study the relations between B and the Bohr
topology of G for some well known groups with boundedness (G, B). As an application, we
prove that the Bohr topology of a topological group which is topologically isomorphic to
the direct product of a locally convex space and an L∞ -group, contains “many” discrete
C-embedded subsets which are C ∗ -embedded in their Bohr compactification. This result
generalizes an analogous theorem of van Douwen for the discrete case and some other ones
due to Hartman and Ryll-Nardzewski concerning the existence of I0 -sets.
We also obtain some results on preservation of compactness for the Bohr topology of
several types of MAP Abelian groups, like L∞ -groups, locally convex vector spaces and
free Abelian topological groups.
1. Introduction. Let G be a maximally almost periodic Abelian
(MAPA) group in the sense of von Neumann. That means, in the Abelian
case, that for all eG 6= x ∈ G there is a continuous character χ such that
χ(x) 6= 1. The class of MAPA groups contains most of the relevant families
of Abelian topological groups. For instance, locally compact Abelian (LCA)
groups, free Abelian topological groups and the additive groups of locally
convex vector spaces are MAPA groups.
Let G be an arbitrary Abelian topological group. The set of all continuous
characters of G, with addition defined pointwise, is an Abelian group again.
b On G two
This group is called the dual group of G and it is denoted by G.
topologies will usually be considered: its original topology and the topology
b The latter is usually called the
of pointwise convergence on elements of G.
+
Bohr topology of G and is denoted by G . The topology of G+ is a totally
1991 Mathematics Subject Classification: 22A05, 22B05, 46A04, 46A99, 54C45, 54H11.
Key words and phrases: Bohr topology, LCA group, L∞ -group, boundedness, locally
convex vector space, DF -space, maximally almost periodic, respects compactness, Cembedded, C ∗ -embedded.
Research partially supported by Spanish DGES grant number PB96-1075.
[195]
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J. Galindo and S. Hernández
bounded group topology, weaker than the original topology of G, and the
completion of G+ is bG, the Bohr compactification of G.
One main problem concerning the Bohr compactification of a topological
group is the identification of those properties of G and G+ that are common
to both topologies. It is not known, in general, how the topology of G+ is
placed with respect to the topology of G. Our aim in this paper is to confront
the families of bounded sets that appear naturally in both G and G+ . A basic
question we are concerned with is when G and G+ share the same compact
subsets. Results in this direction are commonplace in functional analysis,
the Banach–Steinhaus theorem being a good example.
More or less, it can be said that the preservation of compact-like properties from G+ to G involves “uniform boundedness” results and, in many
cases, it can be applied to prove the continuity of certain related algebraic
homomorphisms.
This was the reason which led Glicksberg to prove the following theorem:
Theorem 1.1 [10]. Let G be a LCA group. If K is a subset of G which
is relatively compact in G+ , then K is relatively compact in G.
As Glicksberg himself points out in [10], ‘regarding boundedness in that
situation as meaning “relatively compact”, this theorem is the exact analogue of the uniform boundedness principle for Banach spaces’.
Approaching the problem from a different viewpoint, E. van Douwen [6]
considered the Bohr topology of a discrete Abelian group. A subset A of a
topological space X is said to be C-embedded (respectively C ∗ -embedded ) if
every continuous real-valued function on A (respectively every bounded continuous real-valued function on A) can be extended to a continuous function
on X. Then van Douwen’s main result can be stated in the following way:
Theorem 1.2 [6, Theorem 1.1.3]. Let G be a discrete Abelian group and
let A be an infinite subset of G. Then there is a subset B of A with |B| = |A|
such that B is discrete and C ∗ -embedded in bG and C-embedded in G+ .
In the same paper, van Douwen stated an extension of this theorem
to the real line, but left unanswered whether or not the result is true for
LCA groups. On this line, Hartman and Ryll-Nardzewski [11] had already
obtained the main steps to prove the existence of discrete and C ∗ -embedded
subsets of bG (I0 -sets in their terminology) contained in G, for every LCA
group G.
Van Douwen also applies his theorem to prove that the only convergent
sequences in G+
d are the eventually constant ones. This fact suggests that
a generalization of van Douwen’s theorem, which we think is also an interesting goal in itself, will also be a useful tool to handle the generalization of
Glicksberg’s theorem to bounded subsets of MAPA groups. Therefore, the
following two questions are dealt with in this paper.
Boundedness and Bohr compactification
197
Q1. Generalize van Douwen’s result to the class of LCA groups and
other relevant classes of MAPA groups.
Q2. Extend Glicksberg’s result to general MAPA groups and standard
families of bounded subsets in them.
To investigate these two questions, the notion of boundedness will be
basic. It appears in Chapter 4 of [1] and was introduced by Vilenkin in 1951
in [32] to extend the Pontryagin–van Kampen duality.
To finish this introduction, a few words about notation and terminology.
Let G be an Abelian group. The order of g ∈ G is denoted by o(g). For
A ⊆ G, hAi denotes the smallest subgroup of G containing A. If G is a
topological group, then N0 (G) is the set of all open neighbourhoods
Q of the
identity. If {Gi }i∈I is a collection
of
Abelian
topological
groups,
i∈I Gi is
L
their direct product, and i∈I
G
their
direct
sum
(weak
direct
product).
It
i
Q
will usually
be
assumed
that
G
is
endowed
with
the
Tikhonov
topology
i
i∈I
L
and that
with
the finest group topology making the
i∈I Gi is endowed
L
canonical inclusion Gj → i∈I Gi continuous for every j ∈ I. By the strict
inductive limit of a sequence {G1 ⊆S
. . . ⊆ Gn ⊆ . . .} of Abelian topological
groups will be meant the group G = n<ω Gn endowed with the finest group
topology making continuous all the inclusion mappings of Gn into G.
2. Boundedness. Let G be an Abelian group. According to Vilenkin,
a boundedness on G is a family B of subsets of G, called bounded sets,
satisfying the following conditions:
1.
2.
3.
4.
If X is bounded then so is −X.
Subsets of bounded sets are bounded.
If X and Y are bounded then so are X ∪ Y and X + Y .
Finite sets are bounded.
If a group G is endowed with a boundedness B, then (G, B) is called a
group with boundedness.
Examples of groups with boundedness are: the family of all finite subsets
of any Abelian group; the family of all precompact subsets of a topological
group; the family of all bounded sets (in the usual sense) of the additive
group of a topological vector space.
In [13], Hejcman associated with every topological group G the following
boundedness B: a subset B of G is bounded (i.e., B ∈ B) if and only if for
every neighbourhood U of the identity in G, there exist a finite F ⊂ B and
a natural number n such that
B ⊆ F + U + . n. . + U.
In the sequel it will be assumed that this is the canonical boundedness for
every group under consideration. Notice that (as stated by Hejcman himself
198
J. Galindo and S. Hernández
in [13]) this boundedness coincides with the boundedness of precompact
subsets for LCA groups, and with the usual boundedness for locally convex
spaces.
The concept of boundedness is inherited by subgroups, quotients and
direct products and sums in a natural way:
If {(Gi , Bi ) : i ∈ I} is a family of groups with boundedness, then their
direct product
(resp. direct
sum) G is endowed with the following boundQ
L
edness i∈I Bi (resp. i∈I Bi ): A subset X of G is bounded if πi (X) ∈ Bi
for all i ∈ I (and resp. πi (X) = {0} for all but finitely many i ∈ I), where
πi : G → Gi is the projection.
Given a group with boundedness (G, B) and a subgroup H of G, the
boundedness induced by (G, B) on H is B ∩H. If H is a closed subgroup of G
and π : G → G/H is the canonical quotient mapping, then the boundedness
induced by (G, B) on G/H is π(B).
The following definitions, that stem from the theory of rings of continuous functions, are introduced in connection with the notion of boundedness.
For a group with boundedness (G, B) and a subset A of G, the boundedcovering number , B(A), of A is defined by
n
o
[
F and each F ∈ F is bounded .
B(A) = min |F| : A =
F ∈F
Two subsets A and B of a topological group G are said to be separated
if there are two disjoint closed intervals in T, say I0 and I1 , and a character
b such that χ(A) ⊆ I0 and χ(B) ⊆ I1 .
χ∈G
On the other hand, a boundedness B on G is said to be separated if for
every subset A ⊆ G with bounded-covering number α, there is a subset B of
A with |B| = α such that every subset of B is separated from its complement
in B. We then also say that the group with boundedness (G, B) is separated.
Next, some useful properties of groups with separated boundedness are
stated. The easy proof of the first two of them is omitted.
Proposition 2.1. Let (G, B1 ) and (H, B2 ) be two groups with boundedness such that B2 is separated and let α : G → H be a group homomorphism.
If A is a subset of G with B(A) ≤ B(α(A)), then there exists a subset B of A
such that |B| = B(A) and any subset of B is separated from its complement
in B.
Proposition 2.2. Let {(Gi , Bi ) : i ∈ I} be a family
Q of groups with boundedness
such that every Bi is separated. Consider on Qi∈I Gi the boundedness
Q
B
If either Q
I is finite or B( i∈I Gi ) = ℵ0 , then the
i∈I i described above. Q
group with boundedness ( i∈I Gi , i∈I Bi ) is separated.
Proposition 2.3. Let {(Gi , Bi ) : i ∈ I} be a family of MAPA groups
with boundedness such that every Bi is separated. Suppose in addition that
Boundedness and Bohr compactification
199
B(G
ℵ0 for all i ∈ I. Then the group with boundedness (G, B) =
L i) ≤ L
( i∈I Gi , i∈I Bi ) is separated.
P r o o f. Let A be a subset of G with B(A) ≥ ℵ0 . We have to find B ⊆ A
with |B| = B(A) such that every subset of B is separated from its complement in B.
L
If B(πi0 (A)) = B(A) for some i0 in I (here πi0 : i∈I Gi → Gi0 denotes
the canonical projection), then we can apply Proposition 2.1. Hence, assume
that B(πi (A)) < B(A) for all i ∈ I.
∈ G and C ⊆ G, define supp x = {i ∈ I : πi (x) 6= 0} and
For x S
supp C = {supp x : x ∈ C}. By elementary properties of cardinal numbers
and the definition of bounded-covering number, B(A) = |supp A|. Set now
J0 = {i ∈ I : there exists x ∈ A with i ∈ supp x and o(πi (x)) 6= 2}.
Define
J=
J0
supp A \ J0
if |J0 | = |supp A| = B(A) (case (1)),
if |J0 | < |supp A| (case (2)).
Now we construct by transfinite induction a subset B of A all of whose
subsets are separated by characters from their respective complements.
To begin with, we take an initial ordinal number α equivalent to |J|;
that is, if W (α) denotes the set of ordinals preceding α, then |W (α)| = |J|
and α is the smallest ordinal with this property. For the ordinal 0 we take
an element of A as follows:
• If (1), take any x0 ∈ A such that there is i0 ∈ supp x0 with o(πi0 (x0 ))
6= 2.
• If (2), take any x0 such that there is i0 ∈ supp x0 \ J0 , therefore
o(πi0 (x0 )) = 2.
Now take β ∈ W (α) and suppose that we have already defined a subset
{xγ : γ < β} ⊆ A such that for every γ < β:
S
• If (1), there is iγ ∈ supp xγ \ {supp
S xδ : δ < γ} with o(πiγ (xγ )) 6= 2.
• If (2), there is iγ ∈ supp xγ \ ( {supp xδ : δ < γ} ∪ J0 ), therefore
o(πiγ (xγ )) = 2.
Since α is an initial ordinal and supp x is a finite subset of I for all x ∈ G,
it follows that [
{supp xγ : γ < β} < |W (α)| = |J| = B(A).
Thus, we can find an index iβ ∈ J satisfying:
S
• If (1), iβ ∈ J \ {supp xγ : γ < β}; that is, there is xβ ∈ A with
iβ ∈ supp xβ and o(πiS
(xβ )) 6= 2.
β
• If (2), iβ ∈ J \ ( {supp xγ : γ < β} ∪ J0 ); that is, iβ 6∈ J0 and there is
xβ ∈ A with iβ ∈ supp xβ and, as a consequence, o(πiβ (xβ )) = 2.
200
J. Galindo and S. Hernández
Thus, by transfinite induction, we obtain a subset B = {xβ : β < α}
such that |B| = B(A) and every xβ satisfies:
S
• If (1), there is iβ ∈ supp xβ \ {supp xγ : γ < β} with o(πiβ (xβ )) 6= 2.
S
• If (2), there is iβ ∈ supp xβ \ ( {supp xγ : γ < β} ∪ J0 ). Therefore,
o(πiβ (xβ )) = 2.
Finally, it remains to prove that for every B1 ⊆ B, the sets B1 and B \B1
are separated by characters.
Indeed, let Ia and Ib be two disjoint intervals in the one-dimensional
torus T that are centered at −1 and 1 respectively, and have length greater
than or equal to one third of the total length of T.
b i : γ ∈ W (α)} such that
We shall define inductively a family {χiγ ∈ G
γ
Y
(∗)
χiδ (πiδ (xγ )) ∈ Iγ for all γ < α.
δ≤γ
To simplify notation, set J β = {δ : δ < β and iδ ∈ supp xβ } for every
β < α. Also, let Iβ denote either Ia or Ib depending on whether xβ belongs
to B1 or not.
The first inductive step is simple:
• If (1), consider x0 ∈ B and i0 ∈ supp x0 as defined previously, that is,
with o(πi0 (x0 )) 6= 2. Then it is easily verified (note that the groups under
b i such that χi0 (πi (x0 )) ∈ I0 .
consideration are MAPA) that there is χi0 ∈ G
0
0
bi
• If (2), take x0 ∈ B and i0 ∈ supp x0 \ J0 . Again, there is χi0 ∈ G
0
i0
such that χ (πi0 (x0 )) ∈ I0 .
b i : γ < β} satisfying (∗), for
Suppose we have already defined {χiγ ∈ G
γ
some β in W (α). Since supp xβ is finite, there is γβ < β such that γβ ≥ δ
for any δ ∈ J β . Then
Y
Y
χiδ (πiδ (xβ )) = χiβ (πiβ (xβ )) ·
χiδ (πiδ (xβ )).
δ≤β
Now, consider tβ =
δ≤γβ
Q
δ≤γβ
χiδ (πiδ (xβ )) ∈ T.
• If (1), since the length of t−1
β Iβ is again at least one third of that of
b i such that χiβ (πi (xβ )) ∈ t−1 Iβ .
the torus, it is easy to find χiβ ∈ G
β
β
β
• If (2), since o(πiδ (xβ )) = 2 for every δ < α, tβ is necessarily 1 or
iβ
b i such that
−1. So t−1
∈ G
β
β Iβ is again either Ia or Ib , and there is χ
χiβ (πiβ (xβ )) ∈ t−1
I
.
β
β
In either case condition (∗) holds for β. Hence, we have obtained a family
b i : β ∈ W (α)} satisfying (∗).
{χ ∈ G
β
iβ
Boundedness and Bohr compactification
201
Q
bi ⊆ G
b as
Now, we define a continuous character χ = (χi )i∈I ∈ i∈I G
follows:
0
if i 6= iδ for any δ < α,
χi =
χiδ if i = iδ for some δ < α.
Q
Then χ(xβ ) = δ<α χiδ (πiδ (xβ )) for every β < α. But since iδ does not
belong to supp xβ unless δ ≤ β, we obtain
Y
χiδ (πiδ (xβ )) ∈ Iβ for all β < α
χ(xβ ) =
δ≤β
and the proof is complete.
The following proposition shows that our canonical boundedness is natural for direct products. The easy proof is omitted.
Proposition 2.4. Let {(Gi , Bi )}i∈I be a collection ofQtopological groups
provided with their canonical
boundedness and let G = i∈I Gi . Then the
Q
product boundedness i∈I Bi and the canonical boundedness on G coincide.
3. Groups with separated boundedness. In this section it is proved
that Abelian L∞ -groups and the additive groups of DF -spaces endowed
with their canonical boundednesses are separated.
We recall that an L∞ -group is a topological group whose topology is the
intersection of a non-increasing sequence of locally compact group topologies. This notion was introduced by Varopoulos in [30] where he made a
deep study of these groups, setting the foundations of harmonic analysis on
L∞ -groups. Other relevant contributions are the works of Sulley [23] and
Venkataraman [31].
We also recall that the DF -spaces are a wide class of locally convex
spaces containing normed spaces and duals of metrizable spaces, which is
closed under most common operations like products, quotients, inductive
limits. . . Every DF -space has a fundamental sequence of bounded sets, i.e.,
a sequence B1 ⊂ B2 ⊂ . . . of bounded subsets such that every bounded
subset is contained in some Bk . Thus, if E is a DF -space it is clear that
B(E) = ℵ0 for the canonical boundedness on E. For a precise definition of
DF -spaces and their basic properties the reader may consult, for instance,
[16, §29.3]. For our purposes, we just need to quote that, by [16, §29.3(1)],
the dual space of a DF -space is metrizable and complete.
Let T denote the one-dimensional torus, now identified with the unit
circle of the complex plane. For each p ∈ T, the set {x ∈ T : xn = p} splits
the torus into n disjoint connected components of the same length. Clearly,
the length of each component can be made arbitrarily small by taking n
large enough.
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J. Galindo and S. Hernández
Now, we prove a series of technical lemmas.
b
Lemma 3.1. Let G be a locally connected MAPA group and let A ⊆ G.
Then the following assertions are equivalent:
(a) A is equicontinuous.
S
(b) There is V ∈ N0 (G) such that a∈A a(V ) 6= T.
P r o o f. It is trivial that (a) implies (b).
with p 6∈
S Conversely, assume that there is V ∈ N0 (G) and p ∈ T 2πit
a(V
).
Let
U
be
a
neighbourhood
of
1
in
T
contained
in
{e
: |t| ≤
a∈A
1/4} such that p 6∈ U . Take n large enough so that the connected component of T \ {x ∈ T : xn = p} which contains the identity, Cn (1), lies in
U . Since the mapping n : G → G defined by n(x) = nx is a continuous
homomorphism, there exists a connected W ∈ N0 (G) (note that G is locally
connected) such that n(W ) ⊆ V . It follows that, for all a ∈ A,
a(n(W )) ⊆ a(V ) ⊆ T \ {p}.
Thus, for x ∈ W , a(n(x)) = (a(x))n ∈ T \ {p}. That is, a(W ) ⊆ T \ {x ∈
T : xn = p}. Moreover, 1 ∈ a(W ) for all a ∈ A, and a(W ) is connected.
b :
Hence a(W ) ⊆ Cn (1) ⊆ U for all a ∈ A. This proves that A ⊆ {χ ∈ G
χ(W ) ⊆ U }, which is to say that A is equicontinuous (see for instance [19,
Lemma 2.1]).
The following result generalizes a similar result of van Douwen for the
real line [6, Theorem 9.2].
Lemma 3.2. Let G be a locally connected MAPA group which is either
b is not equicontinuous,
locally compact or completely metrizable. If A ⊆ G
∞
then for any sequence {λk }k=1 of positive numbers there is an infinite subset
∞
B = {χk }∞
k=1 ⊆ A such that for any sequence {Ik }k=1 of intervals in T with
length l(Ik ) = λk , there is x ∈ G with χk (x) ∈ Ik for all k < ω.
P r o o f. First we construct by induction a countable family {Bk }∞
k=1 ⊆
N0 (G) and a sequence {χk }∞
⊆
A
such
that
B
is
symmetric,
χ
(B
)
k
k
k = T,
k=1
l(χk−1 (2Bk )) < λk and 2Bk ⊆ Bk−1 for all k < ω.
Let C be the component of 0 in G. Since A is not equicontinuous and
C is an open subset of G, there must exist χ1 ∈ A such that χ1 (C) 6= {0}.
Since χ1 (C) is a non-trivial connected subgroup of T, χ1 and B1 = G can
be chosen for the inductive step k = 1. Assume now that {B1 , . . . , Bk } and
{χ1 , . . . , χk } satisfying the foregoing conditions have already been found.
Then take Bk+1 ∈ N0 (G) connected
Sand symmetric with l(χk (2Bk+1 )) < λk
and 2Bk+1 ⊆ Bk . By Lemma 3.1, χ∈A χ(Bk+1 ) = T. Take χk+1 ∈ A such
that −1 ∈ χk+1 (Bk+1 ). As Bk+1 is connected and symmetric, it follows that
χk+1 (Bk+1 ) = T. Thus the inductive construction of Bk and χk is complete.
Boundedness and Bohr compactification
203
Let now {Ik }∞
k=1 be a sequence of intervals with l(Ik ) = λk and let tk be
their middle points.
Again inductively, we will find a sequence {xk }∞
k=1 ⊆ G with xk ∈ Bk
such that χk (x1 + . . . + xk ) = tk and χk (x1 + . . . + xk + 2Bk+1 ) ⊆ Ik .
For k = 1, take x1 ∈ B1 = G such that χ1 (x1 ) = t1 .
Suppose that x1 , . . . , xk have been found as required. Since χk+1 (Bk+1 )
= T, we have χk+1 (x1 + . . . + xk + Bk+1 ) = T. Therefore, there exists xk+1 ∈
Bk+1 such that χk+1 (x1 +. . .+xk +xk+1 ) = tk+1 . As Bk+2 is symmetric and
l(χk+1 (2Bk+2 )) ≤ λk+1 , it follows that χk+1 (x1 +. . .+xk+1 +2Bk+2 ) ⊆ Ik+1 .
Finally, let {Ck }∞
k=1 be the family of subsets of G defined by
Ck = x1 + . . . + xk + 2Bk+1 .
It is clear that χk (Ck ) ⊆ Ik and Ck ⊆ Ck−1 for all k < ω. Hence, if every Bk
is chosen either compact, which is possible whenever G is locally compact,
or
and complete, we have
T∞of diameter less than 1/k, in case G is metrizable
T∞
k=1 Ck 6= ∅. It is now enough to take x ∈
k=1 Ck to complete the proof.
Remark 3.3. The preceding lemma can be extended without difficulty to
Čech-complete topological groups, but the present formulation is sufficient
for our purposes.
It is clear that, for non-locally connected groups, Lemma 3.2 cannot always hold. For example if G is a group containing an element x of order
two, then it is impossible to map x homomorphically into any subinterval
of T unless that interval meets 1 or −1. Nevertheless, when G is a compact
group, a variant of Lemma 3.2 is satisfied. Notice that, when G is a comb are the finite ones. Thus,
pact group, the only equicontinuous subsets of G
an application of Pontryagin duality carries out the analogy between the
following lemma and Lemma 3.2.
Lemma 3.4. Let G be a discrete Abelian group and let A be an infinite
subset of G. For every pair I0 , I1 of disjoint closed intervals in T, each
containing at least one nth rooth of unity for all 2 ≤ n < ω, there exists a
b
subset B of A with |B| = |A| such that, for all φ ∈ {0, 1}B , there is χ ∈ G
with χ(b) ∈ Iφ(b) for all b ∈ B.
P r o o f. The idea of the proof is to apply Zorn’s Lemma.
First of all, notice that if there is a subset B1 of A with |B1 | = |A| such
that B1 is contained in a finitely generated subgroup H of G, then there is
m < ω such that H ∼
= Zm × F with F a finite Abelian group. Thus, since
m
the dual group of Z × F is locally connected, Lemma 3.2 can be applied
b to get a subset B of B1 with the required properties.
to B1 and H
Hence, from now on, it will be assumed that subsets of cardinality |A|
are not included in finitely generated subgroups of G.
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J. Galindo and S. Hernández
b : χ(b) ∈
Now, for B ⊆ A and φ ∈ {0, 1}B define Nφ (B) := {χ ∈ G
Iφ(b) ∀b ∈ B}. Notice that Nφ (B) is always compact. Set
A := {B ⊆ A : Nφ (B) 6= ∅ for all φ ∈ {0, 1}B }.
The family A is not empty. Indeed, take x ∈ A and φ ∈ {0, 1}{x} . If o(x) =
∞, define χ(x) arbitrarily in Iφ(x) and then extend it to a homomorphism
of G into T. If o(x) = n, then there exists t ∈ Iφ(x) such that tn = 1. Then
define χ(x) = t and extend it to a homomorphism of G into T. In either
case, the extended homomorphism belongs to Nφ ({x}). Therefore A 6= ∅.
The set A can be ordered in the following way: B1 B2 iff
1. B1 ⊆ B2 and
2. (Nφ (B2 ))|B1 = (Nφ|B1 (B1 ))|B1 for all φ ∈ {0, 1}B2 .
The set A with this order is S
inductive. Indeed, assume that {Bα : α ∈ Λ} is
B
a chain in A and
define
B
=
α∈Λ Bα . Given φ ∈ {0, 1} it is easily verified
T
that Nφ (B) = α∈Λ Nφ|Bα (Bα ). Thus, it is enough to show that the family
{Nφ|Bα (Bα ) : α ∈ Λ} has the finite intersection property. Take α1 , . . . , αn
in Λ with Bα1 Tn. . . Bαn . From the way the order has been defined in A,
it is clear that i=1 Nφ|Bα (Bαi ) = Nφ|Bαn (Bαn ) 6= ∅. Hence B ∈ A and it
i
remains to check that B is an upper bound of the chain. Let α0 ∈ Λ and
χ ∈ Nφ|Bα (Bα0 ). For every α ∈ Λ, define
0
Pα = {ψ ∈ Nφ|B (Bα ) : ψ|Bα = χ|Bα }.
α
0
0
Since Λ is a chain, the family {Pα }α∈Λ of compact
T sets has the finite intersection property. Therefore, we can choose ψ ∈ α∈Λ Pα . Then ψ ∈ Nφ (B)
and ψ|Bα = χ|Bα , showing that Bα0 B. Hence, by Zorn’s Lemma, there
0
0
exists a maximal set B in A.
Suppose now that |B| < |A|. By the assumption at the beginning of the
proof, we can assume that |hBi| < |A| and hence that A is not contained in
hBi.
Suppose for the moment that every a ∈ A has order 2 and consider
a ∈ A \ hBi. Set B 0 = B ∪ {a}. Since hB 0 i = hBi ⊕ hai, it can easily be
proved that B 0 ∈ A and B ≺ B 0 , contrary to the maximality of B.
Next we consider the general case. For every x ∈ A \ hBi we set as above
Bx = B ∪ {x}. It suffices to find x such that Bx ∈ A and B ≺ Bx .
Take φ ∈ {0, 1}Bx . Let n0 x (n0 ∈ N ∪ {0}) be a generator of the cyclic
group hxi ∩ hBi. If n0 ≥ 3 the interval Iφ(x) contains at least one n0 th rooth
of every element of T, hence we can take t ∈ Iφ(x) with tn0 = χ(n0 x) and
the same can be done if n0 = 0. By defining χ
e(x) = t, the homomorphism
χ|hBi is extended to χ
e ∈ Nφ (Bx ). This proves that Nφ (Bx ) 6= ∅ and hence
B ≺ Bx0 .
Boundedness and Bohr compactification
205
Suppose finally that 2x ∈ hBi for every x ∈ A. Consider the quotient
group G/hBi and the canonical epimorphism p : G → G/hBi. Every element
of p(A) has order two, thus by the particular case above there is a subset B1
of p(A) with |B1 | = |p(A)| satisfying the desired properties. If we choose for
every b ∈ B1 an element xb ∈ A with p(xb ) = b it is easily checked that these
properties are also satisfied by B0 = {xb ∈ A : b ∈ B1 }. Since |hBi| < |A|,
we have |p(A)| = |A|, hence |B0 | = |p(A)| = |A| and the proof is complete.
In the sequel, the concept of separated boundedness will be studied for
a class of groups containing the additive groups of locally convex vector
spaces and LCA groups. For vector spaces there is a well known relationship
between continuous characters and continuous linear functionals [14, 23.32]:
if G is the additive group of a locally convex vector space E, then for every
b there is a unique continuous linear functional f on E such that χ(x) =
χ∈G
2πif (x)
e
for all x ∈ G. In addition, this relationship defines a continuous
b
isomorphism φE of E 0 onto G.
The following lemma will simplify several proofs in the sequel.
Lemma 3.5. Let G be a MAPA group containing an open subgroup K =
lim
n topologically isomorphic to the strict inductive limit of the sequence
−→ K∞
{Kn }n=1 of compact groups and let A ⊂ K be a subset not contained in Kn
for any n < ω. Then there is a continuous homomorphism
ψ:G→
∞
M
Tn
(each Tn ∼
= T)
n=1
and a subset A1 of A such that,Lfor every infinite subset B ⊂ A1 , ψ(B) is
m
not contained in the finite sum n=1 Tn for any m < ω.
P r o o f. Since each Kn is a compact subgroup of Kn+1 , there is an in∞
creasing sequence {mn }∞
n=1 of positive integers and two sequences {χn }n=1
∞
b
⊆ K and {xn }n=1 ⊆ A such that xn ∈ Kmn +1 \ Kmn , χn (Kmn ) = {1} and
χn (xn ) 6= 1.
L∞
Define ψ : K → n=1 Tn (with Tn ∼
= T for all n < ω) as follows:
ψ(x) = (χ1 (x), χ2 (x), . . .).
This is a well defined group homomorphism, because for all x ∈ K there is
some n < ω with x ∈ Kmn , and consequently χm (x) = 1 for all m ≥ n. Let
us see that ψ is also continuous. Indeed, let in : Kn → K be the inclusion.
Since K = lim
continuous for all n < ω.
−→ Kn , it suffices to prove that ψ ◦ in isL
Qp
p
Choose mp such that n ≤ mp ; then (ψ ◦in )(Kn ) ⊆ j=1 Tj = j=1 Tj , and
Lp
if πj : j=1 Tj → Tj denotes the canonical projection, then πj ◦ ψ ◦ in = χj .
Thus, πj ◦ ψ ◦ in is continuous for all j with 1 ≤ j ≤ p. This implies that
ψ ◦ in is continuous for all n < ω, showing the continuity of ψ.
206
J. Galindo and S. Hernández
L∞
Since
n=1 Tn is a divisible group, the homomorphism ψ can be extended to G and, being continuous on the open subgroup K, the extended
homomorphism is also continuous.
If A1 = {xn }∞
n=1 , it is clear from the definition
Lm of ψ that no infinite
subset of A1 is mapped by ψ into a finite sum n=1 Tn , m < ω.
Theorem 3.6. Let (G, B) be a MAPA group with its canonical boundedness. Suppose that G is topologically isomorphic to a group of the form
E × G0 , where E is a DF -space and G0 contains an open subgroup which
is a strict inductive limit of compact groups. Then (G, B) is a group with
separated boundedness.
P r o o f. By Propositions 2.2 and 2.4, it is enough to show the separability
for each factor E and G0 .
Let A be an unbounded subset of the DF -space E. We have B(E) = ℵ0 ,
as remarked at the beginning of this section. Thus it is required to find a
countably infinite subset B of A such that every subset of B is separated by
characters from its complement in B.
The set A may be considered as a subset of E 00 , the bidual topological
vector space. Hence, we may apply Lemma 3.2 for G being the additive
group of E 0 with its standard topology (of uniform convergence on bounded
subsets of E), which is complete and metrizable. Thus, it is enough to prove
that φE 0 (A) is not equicontinuous as a set of mappings from E 0 to T.
Suppose that φE 0 (A) is equicontinuous. By Lemma 3.1, there exists an
absolutely convex closed neighbourhood V of zero in E 0 and p ∈ T such that
[
[φE 0 (a)](V ) ⊆ T \ {p}.
a∈A
The inclusion above implies that a(V ) is properly included in [0, 1) for all
a ∈ A. Otherwise, a(V ) being symmetric and connected, it would contain
all the interval [−1, 1], and therefore φE 0 (a)(V ) = T. Thus, |ha, f i| ≤ 1 for
all a ∈ A and f ∈ V . This implies that A ⊆ V ◦ (the polar of V ). Hence
clearly A is a bounded subset of E, contrary to assumption. This finishes
the proof for DF -spaces.
Now assume that G is a MAPA group containing an open subgroup which
is a strict inductive limit of a sequence of compact groups, say K = lim
−→ Kn .
Again, we consider an unbounded subset A of G and we look for a subset
B of A such that |B| = B(A) and every subset of B is separated from its
complement in B.
Let p : G → G/K be the quotient mapping. Since G/K is discrete,
B(p(A)) = |p(A)| and Lemma 3.4 applies to p(A). Consequently, if B(A) ≤
|p(A)|, then Proposition 2.1 applied to the map p finishes the proof.
Now assume that |p(A)| < B(A). Since K is a σ-compact subgroup of
G, it follows that |p(A)| is finite and B(A) = ℵ0 . Thus, there exists a coset
Boundedness and Bohr compactification
207
L of K such that B(A ∩ L) = ℵ0 . Take D = A ∩ L; then there exists x0 ∈ G
with D ⊆ x0 + K. We may assume without loss of generality that x0 is
the identity of G. Obviously, D is not contained in Kn forLany n < ω and
∞
Lemma 3.5 yields a continuous homomorphism
ψ : G0 → n=1 Tn so that
Lm
ψ(D) is not contained in any finite sum n=1 Tn for m < ω. It is then easily
verified that B(ψ(D))
L∞= B(D) = ℵ0 . The proof is completed by applying
Proposition 2.3 to n=1 Tn and then Proposition 2.1 to ψ.
Corollary 3.7. Let G be an Abelian L∞ -group. Then the canonical
boundedness on G is separated.
P r o o f. It can be easily deduced from the structure theorems of [31]
and [23] that every Abelian L∞ -group is isomorphic to the direct product
G = E × G0 , where E is the direct sum of countably many copies of the real
line R and G0 contains an open subgroup isomorphic to the strict inductive
limit of a sequence of compact Abelian groups.
By the stability properties of DF -spaces, E is a DF -space. Thus every L∞ -group satisfies the hypothesis of Theorem 3.6 and the corollary
follows.
As mentioned above, it may happen that some groups receive more than
one boundedness by means of the operations of taking subgroups, quotients,
etc., explained in Section 2. In fact, this kind of groups provide examples of
groups with a boundedness that is not separated by continuous characters.
The next example arises from some facts pointed out in [1, pp. 155 and 162].
It is mainly an adaptation of [29, Theorem 12].
Example 3.8. Let D = D(Ω) be the space of test fuctions on Ω, an
open subset of the euclidean space Rn . In [28] it is proved that D contains
a closed subspace Q such that D/Q is topologically isomorphic to a dense
non-closed subspace of Rω , the direct product of countably many copies of
the real line. On the other hand, D is, as a locally convex space, the strict
inductive limit of its closed subspaces Dn = D(Kn ), where {Kn }∞
n=1 is a
covering of Ω formed by compact sets. It is well known [16, §19.5.4] that a
subset of D is bounded if and only if it is contained in Dn0 for some n0 < ω.
Now, D/Q is not complete, hence if p : D → D/Q is the quotient mapping,
then p(Dn ) 6= p(D) and there is a sequence {xk }∞
k=1 ⊆ D/Q and a sequence
{nk }∞
of
positive
integers
such
that
x
∈
p(D
k
nk+1 ) \ p(Dnk ). Since D/Q is
k=1
metrizable, there exists a sequence {λk }∞
of
positive numbers such that
k=1
∞
the set A = {λk xk }k=1 is bounded in D/Q. Obviously, A is not contained in
p(Dn ) for any n < ω, hence A is not bounded in the boundedness induced
by D on D/Q.
Considering D/Q as a subspace of Rω , it follows that A is a bounded
subset of Rω , and hence clRω A is a compact subset of Rω . If the quotient
boundedness induced by D on D/Q were separated by continuous characters,
208
J. Galindo and S. Hernández
we could apply Corollary 4.3 (which is proved below) to deduce that A, being
relatively compact, must be bounded, contrary to the choice of A.
4. Respecting compactness. The two questions pointed out in the
introduction will be answered in this section. First, van Douwen’s result for
discrete groups is extended to a wider class of groups containing strictly
the class of LCA groups. This gives an answer to Q1. Then it is proved
how this extension yields two general results concerning the preservation of
compactness from the Bohr topology to the original topology of some MAPA
groups. This extends Glicksberg’s results and provides an answer to Q2.
First, we prove some results that will be needed in the sequel.
Lemma 4.1. Let (G, B) be a MAPA group with separated boundedness.
For every subset A of G there exists B ⊆ A such that |B| = B(A), B is
discrete in G+ and C ∗ -embedded in bG.
P r o o f. Let A be a subset of G with B(A) ≥ ℵ0 . By hypothesis the
boundedness B is separated, hence there is B ⊆ A with |B| = B(A) such
that every subset of B is separated from its complement in B.
Take b ∈ B and consider the subsets {b} and B \ {b}. There are two
b such that χ(b) ∈ I0 and
disjoint intervals in T, I0 and I1 , and χ ∈ G
−1
χ(B \ {b}) ⊆ I1 . Then χ (T \ I1 ) is a neighbourhood of b in the Bohr
topology of G that meets B exactly in the point b. That is, B is discrete in
the Bohr topology.
To see that B is C ∗ -embedded in bG it is enough to prove that for every
pair of disjoint subsets of B, say B0 and B1 , there is f ∈ C(bG) such that
0 ≤ f ≤ 1, f|B0 = 0 and f|B1 = 1. Since B is separated there are two disjoint
b such that χ(B0 ) ⊆ I0 and χ(B1 ) ⊆ I1 . Let
intervals, I0 , and I1 and χ ∈ G
now g : T → R be defined by
g(t) =
d(t, I0 )
;
d(t, I0 ) + d(t, I1 )
clearly g is continuous, 0 ≤ g ≤ 1, g|I0 = 0 and g|I1 = 1. Since every
continuous character extends continuously to the Bohr compactification, it
can be assumed that χ is defined on bG. Taking f = g ◦ χ completes the
proof.
Theorem 4.2. Let (G, B) be a MAPA group with separated boundedness.
B(A)
If A ⊆ G is unbounded and compact in the Bohr topology, then |A| ≥ 22
.
P r o o f. By Lemma 4.1, there exists B ⊆ A such that |B| = B(A), B is
discrete and C ∗ -embedded in bG. Hence clbG B is homeomorphic to βB, the
Stone–Čech compactification of B. Since A is Bohr-compact, clbG B ⊆ A,
|B|
B(A)
and since B is discrete, |βB| = 22 . Thus, |A| ≥ 22
.
Boundedness and Bohr compactification
209
The next corollary follows immediately from the preceding theorem.
Corollary 4.3. Let (G, B) be a MAPA group with separated boundedness such that |G| ≤ c. Then every relatively compact subset of G with the
Bohr topology is bounded.
Remark 4.4. Note that Glicksberg’s theorem, for the special case of
G = Rn , is clearly a consequence of Corollary 4.3 above.
It is actually easy to get a common approach and a self-contained proof
of Glicksberg’s theorem 1.1 in its full generality and of the well known “uniform boundedness principle” for locally convex spaces using the techniques
presented above (see [7]).
The following lemma is a part of van Douwen’s theorem [6, Thm. 1.1.3],
so we omit its proof here. An alternative proof of van Douwen’s theorem
which is based on the methods of this paper is given in [8].
Lemma 4.5. Let G be a discrete Abelian group and let A be an infinite
subset of G. Then A has a subset B with |B| = |A| which is discrete and
C-embedded in G+ .
A proof of the following lemma can be found in [25].
Lemma 4.6. If φ : G → H is a continuous homomorphism of topological
groups, then φ : G+ → H + is also continuous with respect to the Bohr
topologies.
Theorem 4.7. Let G be a MAPA group topologically isomorphic to a
group of the form E × G0 , where E is a DF -space and G0 contains an open
subgroup which is the strict inductive limit of a sequence of compact groups.
Endow G with its canonical boundedness. Then, for every subset A ⊆ G,
there exists B ⊆ A with |B| = B(A) such that B is relatively discrete in
G+ , C ∗ -embedded in bG and C-embedded in G+ .
P r o o f. Let π1 : G → E and π2 : G → G0 be the canonical projections.
Suppose first that B(A) = B(π1 (A)) = ℵ0 . Since E is a DF -space there
is a sequence {Bn }∞
n=1 of bounded subsets such that every bounded subset
of E is contained in some Bn . For every n < ω there exists xn ∈ π1 (A) \ Bn .
The set C = {xn : n < ω} is a countable subset of π1 (A) with the property
that every infinite subset of C has infinite bounded-covering number. The
set C cannot be weakly bounded since it is not bounded. Thus, there is
φ ∈ E 0 such that φ(C) is an unbounded subset of the real line.
Therefore there exists a subset D = {yn }∞
n=1 ⊆ C such that |φ(yn+1 )| >
|φ(yn )|+1 for every n < ω. Since, by Lemma 4.6, φ : E + → R+ is continuous
it is clear that D is a discrete subset of E + , and it is to be shown that it is
also C-embedded in E + . Indeed, let Vn = {x ∈ E : |φ(x) − φ(yn )| < 1/2}
210
J. Galindo and S. Hernández
and, for every n < ω, take fn ∈ C(E + ) with 0 ≤ fnP≤ 1, fn (yn ) = 1 and
fn (E \ Vn ) = 0. For any g ∈ C(D), the function ge = n<ω g · fn provides a
continuous extension of g to the whole E + .
On the other hand, the group E with its canonical boundedness is separated (see Theorem 3.6). Lemma 4.1 then implies that there is an infinite subset B1 ⊆ D which is C ∗ -embedded in bE. Since B1 ⊆ π1 (A), for
every x ∈ B1 we can choose y(x) ∈ G0 such that (x, y(x)) ∈ A. Then
B = {(x, y(x)) : x ∈ B1 } is a discrete subset of G+ that is C-embedded in
G+ and C ∗ -embedded in bG.
Assume now that B(A) > B(π1 (A)). Let K = lim
sub−→ Kn be the open
∞
group of G0 which is the strict inductive limit of the sequence {Kn }n=1 of
compact groups, and let p : G0 → G0 /K be the quotient mapping.
Assume that B(A) > |(p◦π2 )(A)|. Since K is a σ-compact open subgroup
of G0 it follows that B(A) = B(π2 (A)) = ℵ0 and π2 (A) meets only finitely
many cosets of K. Thus, there must be x0 in G0 such that
((x0 + K) ∩ π2 (A)) \ (x0 + Kn ) 6= ∅
for all n < ω.
There is no loss of generality in assuming that x0 is the identity, that is, K ∩
π2 (A) is not contained in K
Ln∞for any n < ω. Lemma 3.5 yields a continuous
homomorphism ψ : G0 → n=1 Tn and a subset A1 ⊆ ALsuch that neither
∞
ψ(π2 (A1 )) nor any of its infinite subsets is bounded in n=1 Tn (it is not
difficult to see that bounded
L∞ subsets of direct sums are contained in finite
partial sums). Since | n=1 TL
n | ≤ c, Corollary 4.3 implies that ψ(π2 (A1 ))
∞
is not relatively compact in ( n=1 Tn )+ , which is a σ-compact group and,
a fortiori, a realcompact group.
L∞ All these facts imply that ψ(π2 (A1 )) is not
functionally bounded in ( n=1 Tn )+ , and Lemma 4.6 shows that π2 (A1 ) is
not functionally bounded in (G0 )+ either. Now the proof follows exactly as
in the case of E with E + replaced by G+
0.
It remains to handle the case where B(A) = |(p ◦ π2 )(A)| ≥ ℵ0 .
Set C = (p ◦ π2 )(A) ⊆ G0 /K. Since C is an infinite subset of the discrete
group G0 /K, Lemma 4.5 yields a subset C1 ⊆ C with |C1 | = |C| which is
discrete and C-embedded in (G0 /K)+ . The canonical boundedness of G0 /K
is separated by characters, thus there is a subset B1 ⊆ C1 with |B1 | = |C1 |
which is C ∗ -embedded in b(G0 /K). Clearly B1 is also C-embedded in G+
0 /K.
Now, for every z ∈ B1 take exactly one x(z) ∈ E and y(z) ∈ G0 such that
(π2 ◦ p)(x(z), y(z)) = z and define B = {(x(z), y(z)) ∈ G : z ∈ B1 }. It is
easily verified that B has the required properties.
Corollary 4.8. Let G be an Abelian L∞ -group provided with its canonical boundedness. Then every subset A of G has a subset B with |B| = B(A)
such that B is relatively discrete and C-embedded in G+ and C ∗ -embedded
in bG.
Boundedness and Bohr compactification
211
P r o o f. By the structure theorems mentioned
in the proof of Corollary
Lα
3.7, G is topologically isomorphic to ( n=1 R) × G0 , where α ≤ ℵ0 and
G0 contains an open subgroup isomorphic to the strict inductive limit of a
sequence of compact Abelian groups. By Theorem 4.7, the result follows.
The preservation of compactness from the Bohr topology of a MAP group
(generalizing, therefore, Glicksberg’s theorem) has been the object of thorough research, both in Abelian and non-Abelian groups (see [2, 15, 18, 23]).
Comfort, Trigos-Arrieta and Wu [5] made a main contribution by proving
Theorem 4.9 (Comfort, Trigos-Arrieta and Wu). Let G be a LCA group
and let N be a closed metrizable subgroup of bG. For every subset A of G, if
{a + N : a ∈ A} is a compact subset of bG/N , then A + (N ∩ G) is compact
in G.
Motivated by this result, the authors of [5] defined a MAPA group that
strongly respects compactness to be a group which satisfies the conclusion of
Theorem 4.9. They also raised the question of characterizing such groups.
The techniques developed above will now be applied to investigate this
question.
First, we give an example of a group respecting compactness but which
does not strongly respect compactness.
Example 4.10. Let G be a non-closed dense subgroup of T and let N be a
closed subgroup of T such that N ∩G = {1}. Clearly G respects compactness
and N is metrizable. If we consider a sequence {bn }∞
n=1 in G converging to
some x in N , x 6= 1, then B = {bn : n < ω} ∪ {1} is a non-compact subset
of G with B + N compact in bG = T. Thus, G does not strongly respect
compactness.
Now, a series of preliminary lemmas will be proved.
Lemma 4.11. Let G be a MAPA group, A a subset of G, and N a subset
of bG containing the neutral element such that A + N is compact in bG. If
F is an arbitrary subset of A, then there exists A0 ⊆ A with |A0 | ≤ |N | such
that
clbG F ⊆ A0 + N + clG+ (F − F ).
P r o o f. For any x ∈ N such that (A + {x}) ∩ clbG F 6= ∅, we pick exactly
one ax ∈ A with ax + x ∈ clbG F . Then we define
A0 = {ax : x ∈ N and (A + {x}) ∩ clbG F 6= ∅}.
Clearly, A0 ⊂ A and |A0 | ≤ |N |. Take b ∈ clbG F . Since clbG F ⊆ A + N ,
there are a ∈ A and y ∈ N with b = a + y ∈ clbG F . Set b0 = ay + y ∈ clbG F .
Then b−b0 = a−ay ∈ (clbG F −clbG F )∩G, and thus b−b0 ∈ clbG (F −F )∩G =
clG+ (F − F ). Hence b = ay + y + (b − b0 ) ∈ A0 + N + clG+ (F − F ).
212
J. Galindo and S. Hernández
The following lemma permits us to exhibit many MAPA groups that
strongly respect compactness.
Lemma 4.12. Let {(Gi , Bi ) : i ∈ I} be a family of MAPA groups with
separated boundedness such that the cardinality of their Bohr-separable
Q subc
groups is less than 2 . Assume that G is a subgroup of the product i∈I Gi
such that the boundedness on G inherited from the product coincides with
the boundedness of relatively compact subsets. Then G strongly respects compactness.
P r o o f. Consider a subset A of G and a closed subgroup N of bG with
|N | < 2c (surely, this is the case if N is metrizable) such that {a+N : a ∈ A}
is a compact subset of bG/N or, equivalently, A + N is compact in bG.
Suppose that A+(N ∩G) is not compact in G. Being closed, A+(N ∩G)
is not relatively compact in G. ByQhypothesis it cannot be bounded for the
boundedness that G inherits from i∈I Gi . It follows that πi (A+(N ∩G)) =
Ai 6∈ Bi for some i ∈ I.
The boundedness Bi is separated, therefore Lemma 4.1 implies that there
exists Ci ⊆ Ai with |Ci | = B(Ai ) ≥ ℵ0 and Ci relatively discrete in bGi and
C ∗ -embedded in bGi . Let Bi be a countable subset of Ci . Clearly Bi is also
relatively discrete and C ∗ -embedded in bGi . As a consequence, |clbGi Bi | =
|βBi | = 2c .
On the other hand, let πi|G : G → Gi be the restriction of the ith
projection and let bπi : bG → bGi be its continuous extension. It is clear
that bπi (A + N ) = πi (A) + bπi (N ) is compact in bGi , i.e., Ai + bπi (N ) is
compact in bGi . Lemma 4.11 applied to the set Ai + bπi (N ) and the subset
Bi ⊆ Ai shows that there exists A0i with |A0i | ≤ |bπi (N )| such that
clbGi Bi ⊆ A0i + bπi (N ) + clG+ (Bi − Bi ).
i
But clG+ (Bi − Bi ) is contained in clG+ hBi i, which is a Bohr-separable subi
i
group of Gi and, by hypothesis, has cardinality less than 2c . Thus, each
of the subsets in the sum above has cardinality less than 2c , yielding that
|clbGi Bi | < 2c . This contradiction completes the proof.
Corollary 4.13. Let G be a MAPA group whose boundedness of relatively compact subsets is separated. If the cardinality of each Bohr-separable
subgroup of G is less than 2c , then G strongly respects compactness.
The next result is a consequence of Lemma 4.12 for the additive groups
of locally convex spaces. Recall that a locally convex space E is called semiMontel if every bounded subset of E is relatively compact.
Theorem 4.14. Every semi-Montel space strongly respects compactness.
Boundedness and Bohr compactification
213
P r o o f. Every locally convex space E can be embedded as a subspace of
a direct product of Banach spaces [16, §19.9(1)]. It is well known that the
boundedness that E inherits from this product (with norm-boundedness on
each factor) is the canonical boundedness of E which, in the case of a semiMontel space, is exactly the boundedness of relatively compact subsets. Now,
every separable subgroup of a Banach space is a subgroup of a separable
Banach subspace. Closed subspaces of Banach spaces are closed in the Bohr
topology (this is an immediate consequence of the Hahn–Banach theorem)
and the cardinality of a separable Banach space does not exceed c. So,
Lemma 4.12 applies and the result follows.
A locally convex space E is called semireflexive if the canonical linear
mapping from E to E 00 is surjective, and reflexive if this map is a topological
isomorphism.
Remus and Trigos showed in [21] that every locally convex vector space
with weak topology respects compactness. They also characterized the reflexive locally convex spaces that respect compactness as those which are
Montel spaces. Since every semireflexive space is a semi-Montel space when
considered with its weak topology, and since every Montel space is a semiMontel space [16, §23.3(2)], with the aid of Theorem 4.14 we can push the
Remus and Trigos-Arrieta results a bit further.
Corollary 4.15. Let E be a semireflexive locally convex vector space.
Then the following assertions are equivalent:
1. E repects compactness.
2. E strongly respects compactness.
3. E is a semi-Montel space.
Next, we prove one of the main results of this section. It generalizes
Glicksberg’s theorem, Theorem 2.10 of [5] and the main result of [23]. Notice
that, since every L∞ -group is nuclear, it also follows directly from [2] that
L∞ -groups respect compactness.
Theorem 4.16. Every L∞ -group strongly respects compactness.
P r o o f. By the already mentioned structure theorems, G = E × G0 ,
where G0 contains an open subgroup which is the inductive limit
Lof a sequence of compact groups and E is topologically isomorphic to
n≤α Rn
with α ≤ ℵ0 and each Rn topologically isomorphic to the real line.
Let N be a closed subgroup of bG of cardinality less than 2c , and let
A be a subset of G such that {a + N : a ∈ A} is compact in bG/N . Let
π1 : G → E and π2 : G → G0 be the projections, and consider their
extensions bπ1 : bG → bE and bπ2 : bG → bG0 . Since A + (N ∩ G) is closed
in G, it is enough to prove that π1 (A + (N ∩ G)) and π2 (A + (N ∩ G)) are
relatively compact.
214
J. Galindo and S. Hernández
Since bπ1 (A + N ) is compact in bE, Theorem 4.14 shows that π1 (A) +
(bπ1 (N ) ∩ E) is compact in E. Consequently, π1 (A + (N ∩ G)) is relatively
compact in E.
Let K = lim
−→ Kn be the relevant open subgroup of G0 and let p : G0 →
G0 /K and bp : bG0 → b(G0 /K) be the quotient mapping and its extension.
Let B = p(π2 (A + (N ∩ G))). If B is an infinite subset of the discrete
group G0 /K, then van Douwen’s theorem [6, Thm. 1.1.3] or Theorem 4.7
yield a subset C ⊆ B with |C| = |B| which is discrete in (G0 /K)+ and
C ∗ -embedded in b(G0 /K). Once again, it follows that
|clb(G0 /K) C| = 22
|C|
,
whence
|B|
|clb(G0 /K) B| = 22 .
Now B ⊆ bp(bπ2 (A+N )) and we know that A+N is compact in bG. It follows
|B|
that clb(G0 /K) B ⊆ bp(bπ2 (A+N )). Consequently, 22 ≤ |bp(bπ2 (A+N ))| ≤
|p(π2 (A)) + bp(bπ2 (N ))| ≤ max(|B|, |N |), which is impossible.
We thus conclude that p(π2 (A+(N ∩G))) is finite and it suffices to prove
that D = π2 (A + (N ∩ G)) is relatively compact. If D is not compact we
can find x0 ∈ G0 and an increasing sequence {nk }∞
k=1 of integers such that
D ∩ (x0 + Knk+1 ) \ (x0 + Knk ) 6= ∅ for all k < ω.
There is no loss of generality (replace A by A − x0 ) in assuming that x0
is the identity of G. Then D ∩ (K \ Kn ) 6= ∅L
for all n < ω.
∞
Lemma 3.5 gives a mapping ψL: G0 →
n=1 Tn so that ψ(D ∩ K) is
m
T
.
Let
φ = ψ ◦ π2 and let bφ :
not contained
in
any
finite
sum
n
n=1
L∞
[30, Proposition 2, p. 474],
bG → b( n=1 Tn ) be its Bohr extension. ByL
∞
φ(A + (N ∩ G)) L
is not relatively compact in
n=1 Tn . This implies that
∞
φ(A) + [bφ(N ) ∩ n=1 Tn ] cannot be relatively compact either. L
∞
On the other L
hand, φ(A)+bφ(N ) = bφ(A+N ) is compact in b( n=1 Tn ).
∞
of relatively comNow, the group n=1 Tn provided with the boundedness
L∞
pact subsets is separated
(Theorem 3.6) and | n=1 Tn | < 2c . Corollary
L∞
4.13 shows that
n=1 Tn strongly respects compactness. Thus we have
reached
a
contradiction
with the non-relative compactness of φ(A)+[bφ(N )∩
L∞
T
].
This
shows
that
D = π2 (A+(N ∩G)) must be relatively compact
n=1 n
and completes the proof.
As far as we know, there is no result concerning preservation of compactness for free Abelian groups. Moreover, there has been little progress in
extending the Pontryagin–van Kampen duality theory to these groups (see
[17] and [20] for most recent contributions). Our techniques allow handling
free Abelian groups. First, some definitions and notation must be introduced.
A topological space X is called a µ-space if every functionally bounded
subset is relatively compact. For X a completely regular space, L(X) de-
Boundedness and Bohr compactification
215
notes the free locally convex space associated with X, and A(X) is the free
Abelian group on X. Algebraically, L(X) and A(X) can be seen as the sets
of formal linear combinations of elements of X with coefficients in R and Z
respectively. So, A(X) can be algebraically embedded in L(X). It is proved
in [27] (completing an earlier proof of Tkachenko [24]) that this embedding
is also topological.
The following lemma is an elementary result about topological groups.
Lemma 4.17. Let G be a topological group and X a compact subset of G.
Then the subset hXin consisting of all elements in hXi obtained by summing
at most n elements of X ∪ X −1 is compact in G.
Lemma 4.18. If X is a µ-space, then every weakly bounded subset of
L(X) which is also contained in A(X) is relatively compact.
P r o o f. Let B ⊆ A(X) ⊆ L(X) be a weakly bounded subset of L(X).
By Corollary 2 of [26], there is M > 0 and a functionally bounded subset A
of X such that
o
nX
X
|λx | ≤ M, λx = 0 for all x 6∈ A .
B⊆
λx x ∈ L(X) :
x∈X
x
Now, since X is a µ-space, clX A is a compact subset P
and B is contained
in hclX Ai ⊆ A(X). Furthermore, if b ∈ B and b =
x∈A λx x, we have
P
|λ
|
≤
M
,
and
since
each
λ
is
in
Z,
it
follows
that
b ∈ hclX Ai[M ]+1
x
x
x∈A
where [M ] denotes the integer part of M . Thus, B ⊆ hclX Ai[M ]+1 , which is
compact by Lemma 4.17. This proves that B is relatively compact.
Theorem 4.19. If X is a µ-space, then A(X) strongly respects compactness.
P r o o f. The free locally convex space L(X) can be embedded, as every
locally convex space, in the direct product of a family of Banach spaces.
As shown in the proof of Lemma 4.14, Bohr-separable subgroups of Banach
spaces have cardinality not exceeding c. By Lemma 4.18 the boundedness
that A(X) inherits as a subgroup of this product (i.e., the boundedness inherited from L(X)) is exactly the boundedness of relatively compact subsets.
Then Lemma 4.12 implies that A(X) strongly respects compactness.
Corollary 4.20. Let X be any completely regular Hausdorff space. Then
A(X) respects compactness.
P r o o f. It is enough to notice that, if µX denotes the smallest subspace
of βX which is a µ-space and contains X, then A(X) is a dense subgroup
of A(µX).
5. Remarks and open questions. As pointed out in the introduction, our approach was suggested in part by the works of Comfort and
216
J. Galindo and S. Hernández
Ross [4] and van Douwen [6]. In fact, the latter contains a long list of
open problems concerning the Bohr topology of Abelian groups that fostered the study of Bohr topologies within general topology. However, results
related to Theorem 1.1.3 and Lemma 9.2 of [6] had already been studied in
commutative harmonic analysis. See for instance the work of Hartman and
Ryll-Nardzewski [11, 12, 22], where the related concept of I0 -set (or set of
Hartman and Ryll-Nardzewski) is introduced.
Concerning Theorem 4.16, as well as Theorems 4.14 and 4.19, note that
the closed subgroup N of the Bohr compactification involved is not just
metrizable, but of cardinality less than 2c . So, in absence of the continuum
hypothesis, the property we prove is somewhat stronger than strong preservation of compactness.
Concerning separated boundedness, we mention three questions which
suggest possible future research.
In Example 4.10 it is shown that there are MAPA groups respecting
compactness but not strongly respecting compactness. Nevertheless, our example is a non-complete group. All complete MAPA groups that we know to
respect compactness also strongly respect compactness. Our first question
is, therefore,
Question 5.1. Are the properties of respecting compactness and strongly respecting compactness equivalent in the realm of complete MAPA
groups?
Lemmas 3.2 and 3.4 have been a basic tool in our determination of MAPA
groups with separated boundedness. It seems clear that Lemma 3.2 admits
generalizations that will permit broadening the scope of Theorem 4.7.
Question 5.2. Extend Lemmas 3.2 and 3.4 to larger families of MAPA
groups.
Finally, we think that an adaptation of the ideas worked out in this
paper to non-Abelian groups could provide useful tools to study the Bohr
topology in that context.
Question 5.3. Find interesting families of non-Abelian groups provided
with separated boundedness.
We are indebted to Professor W. W. Comfort for some helpful suggestions and references. We also wish to thank the referee for his/her constructive report. They have helped us to improve parts of this paper.
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Departamento de Matemáticas
Universidad Jaume I
12071 Castellón, Spain
E-mail: [email protected]
[email protected]
Received 7 February 1997;
in revised form 27 July 1998 and 20 October 1998